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  25 <div class="section">
  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.hypergeometric_dist"></a><a class="link" href="hypergeometric_dist.html" title="Hypergeometric Distribution">Hypergeometric
  28         Distribution</a>
  29 </h4></div></div></div>
  30 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">hypergeometric</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
  31 <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
  32
  33 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
  34           <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
  35 <span class="keyword">class</span> <span class="identifier">hypergeometric_distribution</span><span class="special">;</span>
  36
  37 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">&gt;</span>
  38 <span class="keyword">class</span> <span class="identifier">hypergeometric_distribution</span>
  39 <span class="special">{</span>
  40 <span class="keyword">public</span><span class="special">:</span>
  41    <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
  42    <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
  43    <span class="comment">// Construct:</span>
  44    <span class="identifier">hypergeometric_distribution</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">r</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">N</span><span class="special">);</span>
  45    <span class="comment">// Accessors:</span>
  46    <span class="keyword">unsigned</span> <span class="identifier">total</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  47    <span class="keyword">unsigned</span> <span class="identifier">defective</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  48    <span class="keyword">unsigned</span> <span class="identifier">sample_count</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  49 <span class="special">};</span>
  50
  51 <span class="keyword">typedef</span> <span class="identifier">hypergeometric_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">hypergeometric</span><span class="special">;</span>
  52
  53 <span class="special">}}</span> <span class="comment">// namespaces</span>
  54 </pre>
  55 <p>
  56           The hypergeometric distribution describes the number of "events"
  57           <span class="emphasis"><em>k</em></span> from a sample <span class="emphasis"><em>n</em></span> drawn from
  58           a total population <span class="emphasis"><em>N</em></span> <span class="emphasis"><em>without replacement</em></span>.
  59         </p>
  60 <p>
  61           Imagine we have a sample of <span class="emphasis"><em>N</em></span> objects of which <span class="emphasis"><em>r</em></span>
  62           are "defective" and N-r are "not defective" (the terms
  63           "success/failure" or "red/blue" are also used). If
  64           we sample <span class="emphasis"><em>n</em></span> items <span class="emphasis"><em>without replacement</em></span>
  65           then what is the probability that exactly <span class="emphasis"><em>k</em></span> items
  66           in the sample are defective? The answer is given by the pdf of the hypergeometric
  67           distribution <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">k</span><span class="special">;</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">n</span><span class="special">,</span>
  68           <span class="identifier">N</span><span class="special">)</span></code>,
  69           whilst the probability of <span class="emphasis"><em>k</em></span> defectives or fewer is
  70           given by F(k; r, n, N), where F(k) is the CDF of the hypergeometric distribution.
  71         </p>
  72 <div class="note"><table border="0" summary="Note">
  73 <tr>
  74 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
  75 <th align="left">Note</th>
  76 </tr>
  77 <tr><td align="left" valign="top"><p>
  78             Unlike almost all of the other distributions in this library, the hypergeometric
  79             distribution is strictly discrete: it can not be extended to real valued
  80             arguments of its parameters or random variable.
  81           </p></td></tr>
  82 </table></div>
  83 <p>
  84           The following graph shows how the distribution changes as the proportion
  85           of "defective" items changes, while keeping the population and
  86           sample sizes constant:
  87         </p>
  88 <p>
  89           <span class="inlinemediaobject"><img src="../../../../graphs/hypergeometric_pdf_1.svg" align="middle"></span>
  90         </p>
  91 <p>
  92           Note that since the distribution is symmetrical in parameters <span class="emphasis"><em>n</em></span>
  93           and <span class="emphasis"><em>r</em></span>, if we change the sample size and keep the population
  94           and proportion "defective" the same then we obtain basically
  95           the same graphs:
  96         </p>
  97 <p>
  98           <span class="inlinemediaobject"><img src="../../../../graphs/hypergeometric_pdf_2.svg" align="middle"></span>
  99         </p>
 100 <h5>
 101 <a name="math_toolkit.dist_ref.dists.hypergeometric_dist.h0"></a>
 102           <span class="phrase"><a name="math_toolkit.dist_ref.dists.hypergeometric_dist.member_functions"></a></span><a class="link" href="hypergeometric_dist.html#math_toolkit.dist_ref.dists.hypergeometric_dist.member_functions">Member
 103           Functions</a>
 104         </h5>
 105 <pre class="programlisting"><span class="identifier">hypergeometric_distribution</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">r</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">N</span><span class="special">);</span>
 106 </pre>
 107 <p>
 108           Constructs a hypergeometric distribution with a population of <span class="emphasis"><em>N</em></span>
 109           objects, of which <span class="emphasis"><em>r</em></span> are defective, and from which
 110           <span class="emphasis"><em>n</em></span> are sampled.
 111         </p>
 112 <pre class="programlisting"><span class="keyword">unsigned</span> <span class="identifier">total</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 113 </pre>
 114 <p>
 115           Returns the total number of objects <span class="emphasis"><em>N</em></span>.
 116         </p>
 117 <pre class="programlisting"><span class="keyword">unsigned</span> <span class="identifier">defective</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 118 </pre>
 119 <p>
 120           Returns the number of objects <span class="emphasis"><em>r</em></span> in population <span class="emphasis"><em>N</em></span>
 121           which are defective.
 122         </p>
 123 <pre class="programlisting"><span class="keyword">unsigned</span> <span class="identifier">sample_count</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 124 </pre>
 125 <p>
 126           Returns the number of objects <span class="emphasis"><em>n</em></span> which are sampled
 127           from the population <span class="emphasis"><em>N</em></span>.
 128         </p>
 129 <h5>
 130 <a name="math_toolkit.dist_ref.dists.hypergeometric_dist.h1"></a>
 131           <span class="phrase"><a name="math_toolkit.dist_ref.dists.hypergeometric_dist.non_member_accessors"></a></span><a class="link" href="hypergeometric_dist.html#math_toolkit.dist_ref.dists.hypergeometric_dist.non_member_accessors">Non-member
 132           Accessors</a>
 133         </h5>
 134 <p>
 135           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 136           functions</a> that are generic to all distributions are supported:
 137           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 138           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 139           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
 140           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
 141           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
 142           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 143           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
 144           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
 145         </p>
 146 <p>
 147           The domain of the random variable is the unsigned integers in the range
 148           [max(0, n + r - N), min(n, r)]. A <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 149           is raised if the random variable is outside this range, or is not an integral
 150           value.
 151         </p>
 152 <div class="caution"><table border="0" summary="Caution">
 153 <tr>
 154 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td>
 155 <th align="left">Caution</th>
 156 </tr>
 157 <tr><td align="left" valign="top">
 158 <p>
 159             The quantile function will by default return an integer result that has
 160             been <span class="emphasis"><em>rounded outwards</em></span>. That is to say lower quantiles
 161             (where the probability is less than 0.5) are rounded downward, and upper
 162             quantiles (where the probability is greater than 0.5) are rounded upwards.
 163             This behaviour ensures that if an X% quantile is requested, then <span class="emphasis"><em>at
 164             least</em></span> the requested coverage will be present in the central
 165             region, and <span class="emphasis"><em>no more than</em></span> the requested coverage
 166             will be present in the tails.
 167           </p>
 168 <p>
 169             This behaviour can be changed so that the quantile functions are rounded
 170             differently using <a class="link" href="../../pol_overview.html" title="Policy Overview">Policies</a>.
 171             It is strongly recommended that you read the tutorial <a class="link" href="../../pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding
 172             Quantiles of Discrete Distributions</a> before using the quantile
 173             function on the Hypergeometric distribution. The <a class="link" href="../../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference
 174             docs</a> describe how to change the rounding policy for these distributions.
 175           </p>
 176 <p>
 177             However, note that the implementation method of the quantile function
 178             always returns an integral value, therefore attempting to use a <a class="link" href="../../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> that requires (or produces) a real valued
 179             result will result in a compile time error.
 180           </p>
 181 </td></tr>
 182 </table></div>
 183 <h5>
 184 <a name="math_toolkit.dist_ref.dists.hypergeometric_dist.h2"></a>
 185           <span class="phrase"><a name="math_toolkit.dist_ref.dists.hypergeometric_dist.accuracy"></a></span><a class="link" href="hypergeometric_dist.html#math_toolkit.dist_ref.dists.hypergeometric_dist.accuracy">Accuracy</a>
 186         </h5>
 187 <p>
 188           For small N such that <code class="computeroutput"><span class="identifier">N</span> <span class="special">&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">max_factorial</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">value</span></code>
 189           then table based lookup of the results gives an accuracy to a few epsilon.
 190           <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">max_factorial</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">value</span></code> is 170 at double or long double
 191           precision.
 192         </p>
 193 <p>
 194           For larger N such that <code class="computeroutput"><span class="identifier">N</span> <span class="special">&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">prime</span><span class="special">(</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">max_prime</span><span class="special">)</span></code> then only basic arithmetic is required
 195           for the calculation and the accuracy is typically &lt; 20 epsilon. This
 196           takes care of N up to 104729.
 197         </p>
 198 <p>
 199           For <code class="computeroutput"><span class="identifier">N</span> <span class="special">&gt;</span>
 200           <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">prime</span><span class="special">(</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">max_prime</span><span class="special">)</span></code>
 201           then accuracy quickly degrades, with 5 or 6 decimal digits being lost for
 202           N = 110000.
 203         </p>
 204 <p>
 205           In general for very large N, the user should expect to lose log<sub>10</sub>N decimal
 206           digits of precision during the calculation, with the results becoming meaningless
 207           for N &gt;= 10<sup>15</sup>.
 208         </p>
 209 <h5>
 210 <a name="math_toolkit.dist_ref.dists.hypergeometric_dist.h3"></a>
 211           <span class="phrase"><a name="math_toolkit.dist_ref.dists.hypergeometric_dist.testing"></a></span><a class="link" href="hypergeometric_dist.html#math_toolkit.dist_ref.dists.hypergeometric_dist.testing">Testing</a>
 212         </h5>
 213 <p>
 214           There are three sets of tests: our implementation is tested against a table
 215           of values produced by Mathematica's implementation of this distribution.
 216           We also sanity check our implementation against some spot values computed
 217           using the online calculator here <a href="http://stattrek.com/Tables/Hypergeometric.aspx" target="_top">http://stattrek.com/Tables/Hypergeometric.aspx</a>.
 218           Finally we test accuracy against some high precision test data using this
 219           implementation and NTL::RR.
 220         </p>
 221 <h5>
 222 <a name="math_toolkit.dist_ref.dists.hypergeometric_dist.h4"></a>
 223           <span class="phrase"><a name="math_toolkit.dist_ref.dists.hypergeometric_dist.implementation"></a></span><a class="link" href="hypergeometric_dist.html#math_toolkit.dist_ref.dists.hypergeometric_dist.implementation">Implementation</a>
 224         </h5>
 225 <p>
 226           The PDF can be calculated directly using the formula:
 227         </p>
 228 <p>
 229           <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric1.svg"></span>
 230         </p>
 231 <p>
 232           However, this can only be used directly when the largest of the factorials
 233           is guaranteed not to overflow the floating point representation used. This
 234           formula is used directly when <code class="computeroutput"><span class="identifier">N</span>
 235           <span class="special">&lt;</span> <span class="identifier">max_factorial</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">value</span></code>
 236           in which case table lookup of the factorials gives a rapid and accurate
 237           implementation method.
 238         </p>
 239 <p>
 240           For larger <span class="emphasis"><em>N</em></span> the method described in "An Accurate
 241           Computation of the Hypergeometric Distribution Function", Trong Wu,
 242           ACM Transactions on Mathematical Software, Vol. 19, No. 1, March 1993,
 243           Pages 33-43 is used. The method relies on the fact that there is an easy
 244           method for factorising a factorial into the product of prime numbers:
 245         </p>
 246 <p>
 247           <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric2.svg"></span>
 248         </p>
 249 <p>
 250           Where p<sub>i</sub> is the i'th prime number, and e<sub>i</sub> is a small positive integer or
 251           zero, which can be calculated via:
 252         </p>
 253 <p>
 254           <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric3.svg"></span>
 255         </p>
 256 <p>
 257           Further we can combine the factorials in the expression for the PDF to
 258           yield the PDF directly as the product of prime numbers:
 259         </p>
 260 <p>
 261           <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric4.svg"></span>
 262         </p>
 263 <p>
 264           With this time the exponents e<sub>i</sub> being either positive, negative or zero.
 265           Indeed such a degree of cancellation occurs in the calculation of the e<sub>i</sub> that
 266           many are zero, and typically most have a magnitude or no more than 1 or
 267           2.
 268         </p>
 269 <p>
 270           Calculation of the product of the primes requires some care to prevent
 271           numerical overflow, we use a novel recursive method which splits the calculation
 272           into a series of sub-products, with a new sub-product started each time
 273           the next multiplication would cause either overflow or underflow. The sub-products
 274           are stored in a linked list on the program stack, and combined in an order
 275           that will guarantee no overflow or unnecessary-underflow once the last
 276           sub-product has been calculated.
 277         </p>
 278 <p>
 279           This method can be used as long as N is smaller than the largest prime
 280           number we have stored in our table of primes (currently 104729). The method
 281           is relatively slow (calculating the exponents requires the most time),
 282           but requires only a small number of arithmetic operations to calculate
 283           the result (indeed there is no shorter method involving only basic arithmetic
 284           once the exponents have been found), the method is therefore much more
 285           accurate than the alternatives.
 286         </p>
 287 <p>
 288           For much larger N, we can calculate the PDF from the factorials using either
 289           lgamma, or by directly combining lanczos approximations to avoid calculating
 290           via logarithms. We use the latter method, as it is usually 1 or 2 decimal
 291           digits more accurate than computing via logarithms with lgamma. However,
 292           in this area where N &gt; 104729, the user should expect to lose around
 293           log<sub>10</sub>N decimal digits during the calculation in the worst case.
 294         </p>
 295 <p>
 296           The CDF and its complement is calculated by directly summing the PDF's.
 297           We start by deciding whether the CDF, or its complement, is likely to be
 298           the smaller of the two and then calculate the PDF at <span class="emphasis"><em>k</em></span>
 299           (or <span class="emphasis"><em>k+1</em></span> if we're calculating the complement) and calculate
 300           successive PDF values via the recurrence relations:
 301         </p>
 302 <p>
 303           <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric5.svg"></span>
 304         </p>
 305 <p>
 306           Until we either reach the end of the distributions domain, or the next
 307           PDF value to be summed would be too small to affect the result.
 308         </p>
 309 <p>
 310           The quantile is calculated in a similar manner to the CDF: we first guess
 311           which end of the distribution we're nearer to, and then sum PDFs starting
 312           from the end of the distribution this time, until we have some value <span class="emphasis"><em>k</em></span>
 313           that gives the required CDF.
 314         </p>
 315 <p>
 316           The median is simply the quantile at 0.5, and the remaining properties
 317           are calculated via:
 318         </p>
 319 <p>
 320           <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric6.svg"></span>
 321         </p>
 322 </div>
 323 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 324 <td align="left"></td>
 325 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 326       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 327       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 328       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 329         Distributed under the Boost Software License, Version 1.0. (See accompanying
 330         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
 331       </p>
 332 </div></td>
 333 </tr></table>
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